This article is from the Lebanon FAQ, by Alaa Dakroub dakroub@leb.net with numerous contributions by others.
If you want to learn Syriac from scratch, and are not interested in all
the misty gritties of grammar, check the book of Malphono Abrohom Nuro,
Suloqo (I believe it is available in the US).
There is also a book from the Kaslik called "Le Syriac pour tout le
Monde". I do not recommend it. But if it is the Only one you can get hold
of, so why not.
I Sweden there are some school books produced: Safro Tobo, by Malphono
Johanon Kashisho. These too are good.
If you are interested in Learning Syriac grammar, then your best bet (if
you can French) is Costaz' "Grammaire Syriac", you can also buy a
dictionary.
There are also some other Grammars for Syriac. Noldeke and Brocklemann,
but these are real heavy stuff.
for those of you who know Hebrew, check S. Muroaka (sp!) Syriac for
Hebraists. A very good book.
There is a basic difference though between these set of books:
The first are to teach "Modern Literary Syriac" this means stuff that is
more for daily use, while the second set is for those who want to dig into
the huge and fascinating Syriac literature from old to the present.
----
An arabic book can be bought in Syria, the title is al-Aramiyah
al-muhkiyah, it teaches spoken Aramaic/Syriac.
 
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